Muffler TL.pdf

7/30/2019 Muffler TL.pdf

1/5

2003-01-1653

A Review of Current Techniques for Measuring Muffler

Transmission Loss

Z. Tao and A. F. Seybert

University of Kentucky

Copyright 2003 Society of Automotive Engineers, Inc.

ABSTRACT

The most common approach for measuring the

transmission loss of a muffler is to determine the incidentpower by decomposition theory and the transmitted power

by the plane wave approximation assuming an anechoic

termination. Unfortunately, it is difficult to construct a fully

anechoic termination. Thus, two alternative measurement

approaches are considered, which do not require an

anechoic termination: the two load method and the two-

source method. Both methods are demonstrated on two

muffler types: (1) a simple expansion chamber and (2) a

double expansion chamber with an internal connecting

tube. For both cases, the measured transmission losses

were compared to those obtained from the boundary

element method. The measured transmission losses

compared well for both cases demonstrating that

transmission losses can be determined reliably without an

anechoic termination. It should be noted that the two-load

method is the easier to employ for measuring

transmission loss. However, the two-source method can

be used to measure both transmission loss and the four-

pole parameters of a muffler.

INTRODUCTION

There are several parameters that describe the acoustic

performance of a muffler and/or its associated piping.

These include the noise reduction (NR), the insertion loss(IL), and the transmission loss (TL). The NR is the sound

pressure level difference across the muffler. Though the

NR can be easily measured, it is not particularly helpful

for muffler design. The IL is the sound pressure level

difference at a point, usually outside the system, without

and with the muffler present. Though the IL is very useful

to industry, it is not so easy to calculate since it depends

not only on the muffler geometry itself but also on the

source impedance and the radiation impedance. The TL

is the difference in the sound power level between the

incident wave entering and the transmitted wave exiting

the muffler when the muffler termination is anechoic; the

TL is a property of the muffler only. The muffler TL may be

calculated from models but is difficult to measure. This

paper will focus on measuring the muffler TL.

The TL can be measured using the decomposition method

[1-4]. The method is based on decomposition theory

which was originally used to measure acoustic properties

in ducts (such as the absorption coefficient and surface

impedance of absorbing materials) [1,5]. If a two-

microphone random-excitation technique is used, the

sound pressure may be decomposed into its incident and

reflected waves. After the wave is decomposed, the sound

power of the input wave may be calculated.

The major drawback of the decomposition method is that

an anechoic termination is required for measuring TL. Inpractice, an anechoic termination could be constructed

using a long exhaust tube, high absorbing materials, horn

shaped pipes or an active sound anechoic termination

However, a fully anechoic termination is difficult to build

particularly one that is effective at low frequencies.

An acoustical element, like a muffler, can also be

modeled via its so-called four-pole parameters [6]

Assuming plane wave propagation at the inlet and outlet

the four-pole method is a means to relate the pressure

and velocity (particle, volume, or mass) at the inlet to tha

at the outlet. Using the four-pole parameters, the

transmission loss of a muffler can also be readily

calculated. Furthermore, if the source impedance is

known, the four-pole parameters of the muffler can be

used to predict the insertion loss of the muffler system

[6].

The experimental determination of the four poles has been

investigated by many researchers. To and Doige [7, 8]

used a transient testing technique to measure the four

pole parameters. However, their results were no

especially good at low frequencies. Lung and Doige [9]

used a similar approach to measure the four-pole


2/5

parameters of uniform tubes, flare tubes and expansion

chambers. They used a two-load, four-microphone

approach, but the method was unstable when the two

loads were not sufficiently different over the entire

frequency range. The most accepted approach today is

the approach developed by Munjal and Doige [10] who

proposed a two-source method for measuring the four-pole

parameters of an acoustic element or combination of

elements. The method can also be used in the presence

of a mean flow.

This paper will compare the decomposition method, the

two-source method and the two-load method. Examples

will include (1) an expansion chamber and (2) a double

expansion chamber with an internal connecting tube. The

methods will be developed mathematically, and then

applied to the examples. The measured results will be

compared with boundary element method (BEM) results.

DECOMPOSITION METHOD

The muffler TL is the acoustical power level difference

between the incident and transmitted waves assuming ananechoic termination [11], i.e.,

,log10 10t

i

W

WTL = (1)

where Wi is the incident sound power and Wt is the

transmitted sound power. Generally, the transmitted

sound power can be easily obtained by simply measuring

the sound pressure at the outlet. The corresponding

sound power can be related to the sound pressure if a

plane wave with no reflection is assumed. However, the

incident sound power is more difficult to measure due tothe sound reflection from the muffler.

As shown in Figure 1, for one-dimensional sound traveling

along a duct, a standing wave develops when a change in

the impedance is encountered at the muffler inlet. The

sound pressure can be decomposed into its incident and

reflected spectra, SAA and SBB, respectively. One way to

decompose the wave is to use the two-microphone

method and to separate the waves using decomposition

theory [4].

By decomposition theory, the auto spectrum of the

incident wave SAA is

,sin4

sin2cos2

12

2

121212122211

kx

kxQkxCSSS

AA

++= (2)

where S11 and S22are the auto spectra of the total acoustic

pressure at points 1 and 2, respectively; C12 and Q12 are

the real and imaginary parts of cross spectrum between

points 1 and 2; k is the wave number; and x12 is the

distance between the two microphones [4].

The rms amplitude of the incident wave sound pressure p

can be founded from

.AAi Sp = (3

It follows that the sound power for each wave can be

expressed in terms of the incident (pi) and transmitted (ptrms pressure amplitudes as

i

i

iS

c

pW

=

2

(4

and

,2

ot

t Sc

pW

= (5

respectively. In Equations (4) and (5), is the fluiddensity, c is the speed of sound, and Siand So are the

muffler inlet and outlet tube areas, respectively. Inserting

Equations (4) and (5) into Equation (1), the TL can be

expressed as

.log10log20 1010o

i

t

i

S

S

p

pTL += (6)

A common error is to attempt to apply decomposition

method to downstream of the muffler using a pair o

microphones if the termination is not anechoic. This wil

not work as pt is not the same as the incident wave

sound pressure downstream.

The TL for the expansion chamber shown in Figure 2 was

measured by the decomposition method. An

anechoic termination whose absorption coefficient isabout 0.95 over 100-3000 Hz frequency range was used in

the measurement. The TL results are compared to

numerical results from the BEM (also shown in Figure 2)

It is apparent that the measured results deviate from the

BEM results over the whole frequency range. This is likely

due to the termination not being anechoic enough.x12

Speaker

Microphones

SAA

SBB

Figure 1 Setup of decomposition theory

1 2 3

Muffler

Anechoic

termination


3/5

TWO-SOURCE METHOD [10]

The two-source method is based on the transfer matrix

approach. An acoustical element can be modeled by its

four-pole parameters, as show in Figure 3. The transfermatrix is

,2

2

1

1

=

v

p

DC

BA

v

p(7)

where p1 and p2are the sound pressure amplitudes at the

inlet and outlet, respectively; v1 and v2 are the particle

velocity amplitudes at the inlet and outlet, respectively;

and A, B, Cand D are the four-pole parameters of the

system.

When using the two-source method, two sound sources

should be placed as shown in Figure 4. Configuration a

will be examined first. Using the transfer matrix method,

one can readily obtain four-pole equations for the straighttube elements between microphones 1-2 and 3-4.

Similarly, the four-pole equation for element 2-3 which

includes the muffler can be expressed as

.3

3

2323

2323

2

2

=

a

a

a

a

v

p

DC

BA

v

p(8)

where the subscript a refers to Configuration a in Figure 4.

Combining the four-pole equations for 1-2, 3-4 and 2-3, the

equation

+

=

aa

a

aa

a

pB

ADCp

B

D

p

DC

BA

pApB

p

4

34

3434343

34

34

3

2323

2323

2121

12

2

)(

)(1

(9)

is developed where Aij, Bij, CijandDij are the four poles for

acoustical element ij; piais the sound pressure and viais

the particle velocity at point i for Configuration a.

One can see in Equation (9) that there are four unknowns

A23,B23, C23 andD23 , but only two equations. By moving

the sound source to the other end (Configuration b in

Figure 4), two additional equations are obtained and the

four poles of element 3-2 can be evaluated. Fo

Configuration b, one can then write

,1

2

2

2323

2323

2

2

1

2323

2323

3

3

=

=

b

b

b

b

b

b

v

p

AC

BD

v

p

DC

BA

v

p(10)

where is the determinant of the matrix, = A23D23-B23C2and minus sign - is from the change of the velocitydirection in Configuration b.

Using the same approach as before, one can obtain

another four-pole equation

=

bb

b

ab

b

pB

Ap

B

DACp

AC

BD

pDpB

p

2

12

121

1212

1212

12

12

2

2323

2323

434434

34

3

)(

1

)(1

(11)

where 12 =A12D12 - B12C12, 34 =A34D34 - B34C34. pib is the

sound pressure and vibis the particle velocity at point i fo

configuration b. Using both Equations (9) and (11), the

four-pole parameters can be written as

)(

)()(

343434

323234343234323423

ab

ababaa

HH

HHDHHHHA

+

= (12)

)(

)(

343434

32323423

ab

ba

HH

HHBB

= (13)

Acoustical element

1

Figure 3 The four-poles

21p

1v

2p

2v Figure 4 Setup of two-source method

1 Muffler 3 42Source

1 3 42 Source

Configuration a

Configuration b

0

10

20

30

40

50

0 500 1000 1500 2000 2500 3000Frequency (Hz)

TL

(dB)

Measured

BEM

Figure 2 Decomposition method vs. BEM

(Muffler dimensions in inches)

6.035

8

1.375 1.375


4/5

)(

((((

34343412

34343432123134343432123123

ab

abbbaa

HHB

DHHAHDHHAHC

=

(14)

,)(

)()(34

34343412

3232123131

23B

HHB

HHAHHD

ab

abba

+

= (15)

where ijij ppH /= , which are measured.

Assuming that flow can be neglected, the four poles for

elements 1-2 and 3-4 can be expressed as

1,cos)/(sin

sincos12

1212

1212

1212

1212 =

=

klcklj

klcjkl

DC

BA(16)

and

1,cos)/(sin

sincos34

3434

3434

3434

3434 =

=

klcklj

klcjkl

DC

BA(17)

respectively. In Equations (16-17), l12 and l34 are the

microphone spacings for elements 1-2 and 3-4,respectively. The TL can then be expressed in terms of

the four-pole parameters and tube areas as [6]

.log102

1log20 102323

232310

+

++

+=o

i

S

SDCc

c

BATL

(18)

It should be noted that the two-source method can be

implemented using only two microphones with random

excitation. One can obtain all necessary transfer functions

Hij by moving one microphone and using the other

microphone as a reference.

The TL comparison is shown in Figure 5 for the expansion

chamber used previously in Figure 2. The two-source

method agreed especially well with the BEM results. The

termination was the straight tube with absorbing material.

Figure 6 shows another TL comparison for a double

expansion chamber. Again, the measured results agreed

with the BEM results.

TWO-LOAD METHOD [9]

In Equation (9), one can see that there are four unknowns

A23, B23, C23 andD23, but there are only two equations.

Instead of moving the sound source to the other end to get

two additional equations, the same result can be obtained

by changing the end condition, as shown in Figure 7.

Changing the end condition effectively changes theimpedance at the termination from Za to Zb. Equations (12

- 15) can be used again, and the four-pole parameters of

element 2-3 can be obtained, as can the TL from Equation

(18).

In the two-load method, it is obvious that if two loads are

very similar, the result will be unstable. Generally, two

loads can be two different length tubes, a single tube with

and without absorbing material, or even two differen

mufflers. In this research, two loads were achieved by a

tube with and without absorbing material.

Figure 8 shows the TL comparison for the double

expansion chamber shown. Excellent agreement was

obtained using the two-load method and the two-source

method.

Figure 9 shows a comparison of the real part of the four

pole parameterA23 for the expansion chamber in Figure 9

Though the TL was measured accurately by the two-load

method, the four-pole parameters are not as clean

0

10

20

30

40

50

0 500 1000 1500 2000 2500 3000

Frequency (Hz)

TL

(dB)

Two-source Method

BEM6.035

8

1.375 1.375

Figure 5 Two-source method vs. BEM


0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000

Frequency (Hz)

TL

(dB)

Two-source MethodBEM

1.375

2.24

8

6.0351.375

4.125

2.24

1.375

Figure 6 Two-source method vs. BEM


1 Test Element 3 42Source

Load 1

Figure 7 Setup of two-load method

1 3 42

Zb

Za

Load 2


5/5

as those measured using the two-source method. This

may be due to the two loads not being sufficiently different

in 500-1500 Hz range. Further investigation is needed to

clarify the reason for these differences.

CONCLUSIONS

Three methods for measuring muffler TL have been

discussed in this paper. The results indicated the

limitations of the decomposition method in the absence of

a good anechoic termination. Furthermore, the

decomposition method does not lead to the four-pole

parameters of the muffler; these are necessary for

predicting the IL of the system. However, both the two-

source and two-load methods accurately measured the

muffler TL without the use of an anechoic termination.Theoretically, any termination could be used, but a

termination with high reflection is not recommended.

When the termination is highly reflective, the signal-to-

noise ratio is low, and random errors can be large, which

may contaminate the experimental results.

The two-load method is easier to employ than the two-

source method, since the sound source does not have to

be moved. This assumes that the two loads are different

enough. However, this study indicated that the two-source

method might be the better choice for determining the

four-pole parameters of a muffler. It is also possible to

reverse the muffler, which may be easier than moving the

sound source.

ACKNOWLEDGMENTS

The authors would like to acknowledge the support of the

Vibro-Acoustics Consortium at the University of Kentucky

REFERENCES

1. Seybert, A.F. and Ross, D.F., Experimenta

Determination of Acoustic Properties Using a Two

microphone Random Excitation Technique, J

Acoust. Soc. Am., 61, 1362-1370 (1977).

2. Chung, J.Y. and Blaser, D.A., Transfer Function

Method of Measuring In-duct Acoustic Properties, I:

Theory, J. Acoust. Soc. Am, 68, 907-913 (1980).

3. Chung, J.Y. and Blaser, D.A., Transfer Function

Method of Measuring In-duct Acoustic Properties, II

Experiment, J. Acoust. Soc. Am, 68, 914-921 (1980)

4. Seybert, A.F., Two-sensor Methods for the

Measurement of Sound Intensity and Acoustic

Properties in Ducts, J. Acoust. Soc. Am, 83, 2233-

2239 (1988).

5. ASTM standard, E1050-98, Standard Test Method for

Impedance and Absorption of Acoustical Materia

Using a Tube, Two Microphones and a Digita

Frequency Analysis System, (1998).

6. Munjal, M.L., Acoustics of Ducts and Mufflers, New

York: Wiley-Interscience (1987).

7. To, C.W.S. and Doige, A.G., A Transient Testing

Technique for The Determination of Matrix Parameters

of Acoustic Systems, 1: Theory and Principles,

Journal of Sound and Vibration, 62, 207-222 (1979).8. To, C.W.S. and Doige, A.G., A Transient Testing

Technique for the Determination of Matrix Parameters

of Acoustic Systems, 2: Experimental Procedures

and Results, Journal of Sound and Vibration, 62, 223

233 (1979).

9. Lung, T.Y. and Doige, A.G., A Time-averaging

Transient Testing Method for Acoustic Properties of

Piping Systems and Mufflers, J. Acoust. Soc. Am,

73, 867-876 (1983).

10. Munjal, M.L. and Doige A.G., Theory of a Two

Source-location Method for Direct Experimenta

Evaluation of the Four-pole Parameters of an

Aeroacoustic Element, Journal of Sound andVibration, 141(2), 323-333 (1990).

11. Beranek, L.L. and Vr, I.L., Noise and Vibration

Control Engineering, John Wiley & Sons, Inc., 374

(1992).

CONTACT

For additional information concerning this article, please

contact Dr. A. F. Seybert at (859) 257-6336 x 80645 or

via email [email protected].

-12

-10

-8

-6

-4

-2

0

2

4

6

0 500 1000 1500 2000 2500 3000

Frequency (Hz)

RealPartofA

Two-source MethodTwo-load Method

6.035

4

1.375 1.375

Figure 9 Real part of the four pole parameter A(Muffler dimensions in inches)

0

20

40

60

80

100

0 500 1000 1500 2000 2500 3000

Frequency (Hz)

TL(

dB)

Two-source Method

Two-load Method1.375

2.24

12

6.0351.375

4.125

2.24

1.375

Figure 8 Two-source method vs. two-load method


Documents

Muffler TL.pdf